Instability and self-contact phenomena in the writhing of clamped rods

We use the Cosserat rod theory to present a unified picture of jump phenomena, associated with looping, snap-through, pop-out, etc., in twisted clamped rods undergoing large deflections. Both contact-free rods and rods with isolated points of self-contact are considered. Taking proper account of the symmetries of the problem we find that an arbitrary contact-free solution is fully characterised by four parameters; each point contact adds another two. A shooting method is used for solving the boundary value problem. An intricate bifurcation picture emerges with a strong interplay between planar and spatial rod configurations. We find new jump phenomena by treating the ratio of torsional to bending stiffness of the rod as a bifurcation parameter. Load-deflection curves are computed and compared with results from carefully conducted experiments on contact-free as well as self-contacting metal-alloy rods.